Theorem canthwdom 8773

Description: Cantor's Theorem, stated using weak dominance (this is actually a stronger statement than canth2 8401, equivalent to canth 6880). (Contributed by Mario Carneiro, 15-May-2015.)

Assertion

Ref	Expression
canthwdom	⊢ ¬ 𝒫 𝐴 ≼* 𝐴

Proof of Theorem canthwdom

Dummy variables 𝑥 𝑓 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		0elpw 5068	. . . . 5 ⊢ ∅ ∈ 𝒫 𝐴
2		ne0i 4149	. . . . 5 ⊢ (∅ ∈ 𝒫 𝐴 → 𝒫 𝐴 ≠ ∅)
3	1, 2	mp1i 13	. . . 4 ⊢ (𝒫 𝐴 ≼* 𝐴 → 𝒫 𝐴 ≠ ∅)
4		brwdomn0 8763	. . . 4 ⊢ (𝒫 𝐴 ≠ ∅ → (𝒫 𝐴 ≼* 𝐴 ↔ ∃𝑓 𝑓:𝐴–onto→𝒫 𝐴))
5	3, 4	syl 17	. . 3 ⊢ (𝒫 𝐴 ≼* 𝐴 → (𝒫 𝐴 ≼* 𝐴 ↔ ∃𝑓 𝑓:𝐴–onto→𝒫 𝐴))
6	5	ibi 259	. 2 ⊢ (𝒫 𝐴 ≼* 𝐴 → ∃𝑓 𝑓:𝐴–onto→𝒫 𝐴)
7		relwdom 8760	. . . . 5 ⊢ Rel ≼*
8	7	brrelex2i 5407	. . . 4 ⊢ (𝒫 𝐴 ≼* 𝐴 → 𝐴 ∈ V)
9		foeq2 6363	. . . . . . 7 ⊢ (𝑥 = 𝐴 → (𝑓:𝑥–onto→𝒫 𝑥 ↔ 𝑓:𝐴–onto→𝒫 𝑥))
10		pweq 4382	. . . . . . . 8 ⊢ (𝑥 = 𝐴 → 𝒫 𝑥 = 𝒫 𝐴)
11		foeq3 6364	. . . . . . . 8 ⊢ (𝒫 𝑥 = 𝒫 𝐴 → (𝑓:𝐴–onto→𝒫 𝑥 ↔ 𝑓:𝐴–onto→𝒫 𝐴))
12	10, 11	syl 17	. . . . . . 7 ⊢ (𝑥 = 𝐴 → (𝑓:𝐴–onto→𝒫 𝑥 ↔ 𝑓:𝐴–onto→𝒫 𝐴))
13	9, 12	bitrd 271	. . . . . 6 ⊢ (𝑥 = 𝐴 → (𝑓:𝑥–onto→𝒫 𝑥 ↔ 𝑓:𝐴–onto→𝒫 𝐴))
14	13	notbid 310	. . . . 5 ⊢ (𝑥 = 𝐴 → (¬ 𝑓:𝑥–onto→𝒫 𝑥 ↔ ¬ 𝑓:𝐴–onto→𝒫 𝐴))
15		vex 3401	. . . . . 6 ⊢ 𝑥 ∈ V
16	15	canth 6880	. . . . 5 ⊢ ¬ 𝑓:𝑥–onto→𝒫 𝑥
17	14, 16	vtoclg 3467	. . . 4 ⊢ (𝐴 ∈ V → ¬ 𝑓:𝐴–onto→𝒫 𝐴)
18	8, 17	syl 17	. . 3 ⊢ (𝒫 𝐴 ≼* 𝐴 → ¬ 𝑓:𝐴–onto→𝒫 𝐴)
19	18	nexdv 1979	. 2 ⊢ (𝒫 𝐴 ≼* 𝐴 → ¬ ∃𝑓 𝑓:𝐴–onto→𝒫 𝐴)
20	6, 19	pm2.65i 186	1 ⊢ ¬ 𝒫 𝐴 ≼* 𝐴

Syntax hints:  ¬ wn 3   ↔ wb 198   = wceq 1601  ∃wex 1823   ∈ wcel 2107   ≠ wne 2969  Vcvv 3398  ∅c0 4141  𝒫 cpw 4379   class class class wbr 4886  –onto→wfo 6133   ≼^* cwdom 8751

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-sep 5017  ax-nul 5025  ax-pr 5138  ax-un 7226

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-ral 3095  df-rex 3096  df-rab 3099  df-v 3400  df-sbc 3653  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4672  df-br 4887  df-opab 4949  df-mpt 4966  df-id 5261  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-iota 6099  df-fun 6137  df-fn 6138  df-f 6139  df-fo 6141  df-fv 6143  df-wdom 8753

This theorem is referenced by:  pwcdadom  9373